// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2012 Giacomo Po <gpo@ucla.edu>
// Copyright (C) 2011-2014 Gael Guennebaud <gael.guennebaud@inria.fr>
// Copyright (C) 2018 David Hyde <dabh@stanford.edu>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.

#ifndef EIGEN_MINRES_H_
#define EIGEN_MINRES_H_

namespace Eigen {

namespace internal {

/** \internal Low-level MINRES algorithm
 * \param mat The matrix A
 * \param rhs The right hand side vector b
 * \param x On input and initial solution, on output the computed solution.
 * \param precond A right preconditioner being able to efficiently solve for an
 *                approximation of Ax=b (regardless of b)
 * \param iters On input the max number of iteration, on output the number of performed iterations.
 * \param tol_error On input the tolerance error, on output an estimation of the relative error.
 */
template<typename MatrixType, typename Rhs, typename Dest, typename Preconditioner>
EIGEN_DONT_INLINE void
minres(const MatrixType& mat,
	   const Rhs& rhs,
	   Dest& x,
	   const Preconditioner& precond,
	   Index& iters,
	   typename Dest::RealScalar& tol_error)
{
	using std::sqrt;
	typedef typename Dest::RealScalar RealScalar;
	typedef typename Dest::Scalar Scalar;
	typedef Matrix<Scalar, Dynamic, 1> VectorType;

	// Check for zero rhs
	const RealScalar rhsNorm2(rhs.squaredNorm());
	if (rhsNorm2 == 0) {
		x.setZero();
		iters = 0;
		tol_error = 0;
		return;
	}

	// initialize
	const Index maxIters(iters);								   // initialize maxIters to iters
	const Index N(mat.cols());									   // the size of the matrix
	const RealScalar threshold2(tol_error * tol_error * rhsNorm2); // convergence threshold (compared to residualNorm2)

	// Initialize preconditioned Lanczos
	VectorType v_old(N);			   // will be initialized inside loop
	VectorType v(VectorType::Zero(N)); // initialize v
	VectorType v_new(rhs - mat * x);   // initialize v_new
	RealScalar residualNorm2(v_new.squaredNorm());
	VectorType w(N);						// will be initialized inside loop
	VectorType w_new(precond.solve(v_new)); // initialize w_new
											//            RealScalar beta; // will be initialized inside loop
	RealScalar beta_new2(v_new.dot(w_new));
	eigen_assert(beta_new2 >= 0.0 && "PRECONDITIONER IS NOT POSITIVE DEFINITE");
	RealScalar beta_new(sqrt(beta_new2));
	const RealScalar beta_one(beta_new);
	// Initialize other variables
	RealScalar c(1.0); // the cosine of the Givens rotation
	RealScalar c_old(1.0);
	RealScalar s(0.0);					   // the sine of the Givens rotation
	RealScalar s_old(0.0);				   // the sine of the Givens rotation
	VectorType p_oold(N);				   // will be initialized in loop
	VectorType p_old(VectorType::Zero(N)); // initialize p_old=0
	VectorType p(p_old);				   // initialize p=0
	RealScalar eta(1.0);

	iters = 0; // reset iters
	while (iters < maxIters) {
		// Preconditioned Lanczos
		/* Note that there are 4 variants on the Lanczos algorithm. These are
		 * described in Paige, C. C. (1972). Computational variants of
		 * the Lanczos method for the eigenproblem. IMA Journal of Applied
		 * Mathematics, 10(3), 373-381. The current implementation corresponds
		 * to the case A(2,7) in the paper. It also corresponds to
		 * algorithm 6.14 in Y. Saad, Iterative Methods for Sparse Linear
		 * Systems, 2003 p.173. For the preconditioned version see
		 * A. Greenbaum, Iterative Methods for Solving Linear Systems, SIAM (1987).
		 */
		const RealScalar beta(beta_new);
		v_old = v;		   // update: at first time step, this makes v_old = 0 so value of beta doesn't matter
		v_new /= beta_new; // overwrite v_new for next iteration
		w_new /= beta_new; // overwrite w_new for next iteration
		v = v_new;		   // update
		w = w_new;		   // update
		v_new.noalias() = mat * w - beta * v_old; // compute v_new
		const RealScalar alpha = v_new.dot(w);
		v_new -= alpha * v;			  // overwrite v_new
		w_new = precond.solve(v_new); // overwrite w_new
		beta_new2 = v_new.dot(w_new); // compute beta_new
		eigen_assert(beta_new2 >= 0.0 && "PRECONDITIONER IS NOT POSITIVE DEFINITE");
		beta_new = sqrt(beta_new2); // compute beta_new

		// Givens rotation
		const RealScalar r2 = s * alpha + c * c_old * beta; // s, s_old, c and c_old are still from previous iteration
		const RealScalar r3 = s_old * beta;					// s, s_old, c and c_old are still from previous iteration
		const RealScalar r1_hat = c * alpha - c_old * s * beta;
		const RealScalar r1 = sqrt(std::pow(r1_hat, 2) + std::pow(beta_new, 2));
		c_old = c;		   // store for next iteration
		s_old = s;		   // store for next iteration
		c = r1_hat / r1;   // new cosine
		s = beta_new / r1; // new sine

		// Update solution
		p_oold = p_old;
		p_old = p;
		p.noalias() = (w - r2 * p_old - r3 * p_oold) / r1; // IS NOALIAS REQUIRED?
		x += beta_one * c * eta * p;

		/* Update the squared residual. Note that this is the estimated residual.
		The real residual |Ax-b|^2 may be slightly larger */
		residualNorm2 *= s * s;

		if (residualNorm2 < threshold2) {
			break;
		}

		eta = -s * eta; // update eta
		iters++;		// increment iteration number (for output purposes)
	}

	/* Compute error. Note that this is the estimated error. The real
	 error |Ax-b|/|b| may be slightly larger */
	tol_error = std::sqrt(residualNorm2 / rhsNorm2);
}

}

template<typename _MatrixType, int _UpLo = Lower, typename _Preconditioner = IdentityPreconditioner>
class MINRES;

namespace internal {

template<typename _MatrixType, int _UpLo, typename _Preconditioner>
struct traits<MINRES<_MatrixType, _UpLo, _Preconditioner>>
{
	typedef _MatrixType MatrixType;
	typedef _Preconditioner Preconditioner;
};

}

/** \ingroup IterativeLinearSolvers_Module
 * \brief A minimal residual solver for sparse symmetric problems
 *
 * This class allows to solve for A.x = b sparse linear problems using the MINRES algorithm
 * of Paige and Saunders (1975). The sparse matrix A must be symmetric (possibly indefinite).
 * The vectors x and b can be either dense or sparse.
 *
 * \tparam _MatrixType the type of the sparse matrix A, can be a dense or a sparse matrix.
 * \tparam _UpLo the triangular part that will be used for the computations. It can be Lower,
 *               Upper, or Lower|Upper in which the full matrix entries will be considered. Default is Lower.
 * \tparam _Preconditioner the type of the preconditioner. Default is DiagonalPreconditioner
 *
 * The maximal number of iterations and tolerance value can be controlled via the setMaxIterations()
 * and setTolerance() methods. The defaults are the size of the problem for the maximal number of iterations
 * and NumTraits<Scalar>::epsilon() for the tolerance.
 *
 * This class can be used as the direct solver classes. Here is a typical usage example:
 * \code
 * int n = 10000;
 * VectorXd x(n), b(n);
 * SparseMatrix<double> A(n,n);
 * // fill A and b
 * MINRES<SparseMatrix<double> > mr;
 * mr.compute(A);
 * x = mr.solve(b);
 * std::cout << "#iterations:     " << mr.iterations() << std::endl;
 * std::cout << "estimated error: " << mr.error()      << std::endl;
 * // update b, and solve again
 * x = mr.solve(b);
 * \endcode
 *
 * By default the iterations start with x=0 as an initial guess of the solution.
 * One can control the start using the solveWithGuess() method.
 *
 * MINRES can also be used in a matrix-free context, see the following \link MatrixfreeSolverExample example \endlink.
 *
 * \sa class ConjugateGradient, BiCGSTAB, SimplicialCholesky, DiagonalPreconditioner, IdentityPreconditioner
 */
template<typename _MatrixType, int _UpLo, typename _Preconditioner>
class MINRES : public IterativeSolverBase<MINRES<_MatrixType, _UpLo, _Preconditioner>>
{

	typedef IterativeSolverBase<MINRES> Base;
	using Base::m_error;
	using Base::m_info;
	using Base::m_isInitialized;
	using Base::m_iterations;
	using Base::matrix;

  public:
	using Base::_solve_impl;
	typedef _MatrixType MatrixType;
	typedef typename MatrixType::Scalar Scalar;
	typedef typename MatrixType::RealScalar RealScalar;
	typedef _Preconditioner Preconditioner;

	enum
	{
		UpLo = _UpLo
	};

  public:
	/** Default constructor. */
	MINRES()
		: Base()
	{
	}

	/** Initialize the solver with matrix \a A for further \c Ax=b solving.
	 *
	 * This constructor is a shortcut for the default constructor followed
	 * by a call to compute().
	 *
	 * \warning this class stores a reference to the matrix A as well as some
	 * precomputed values that depend on it. Therefore, if \a A is changed
	 * this class becomes invalid. Call compute() to update it with the new
	 * matrix A, or modify a copy of A.
	 */
	template<typename MatrixDerived>
	explicit MINRES(const EigenBase<MatrixDerived>& A)
		: Base(A.derived())
	{
	}

	/** Destructor. */
	~MINRES() {}

	/** \internal */
	template<typename Rhs, typename Dest>
	void _solve_vector_with_guess_impl(const Rhs& b, Dest& x) const
	{
		typedef typename Base::MatrixWrapper MatrixWrapper;
		typedef typename Base::ActualMatrixType ActualMatrixType;
		enum
		{
			TransposeInput = (!MatrixWrapper::MatrixFree) && (UpLo == (Lower | Upper)) && (!MatrixType::IsRowMajor) &&
							 (!NumTraits<Scalar>::IsComplex)
		};
		typedef typename internal::conditional<TransposeInput,
											   Transpose<const ActualMatrixType>,
											   ActualMatrixType const&>::type RowMajorWrapper;
		EIGEN_STATIC_ASSERT(EIGEN_IMPLIES(MatrixWrapper::MatrixFree, UpLo == (Lower | Upper)),
							MATRIX_FREE_CONJUGATE_GRADIENT_IS_COMPATIBLE_WITH_UPPER_UNION_LOWER_MODE_ONLY);
		typedef typename internal::conditional<
			UpLo == (Lower | Upper),
			RowMajorWrapper,
			typename MatrixWrapper::template ConstSelfAdjointViewReturnType<UpLo>::Type>::type SelfAdjointWrapper;

		m_iterations = Base::maxIterations();
		m_error = Base::m_tolerance;
		RowMajorWrapper row_mat(matrix());
		internal::minres(SelfAdjointWrapper(row_mat), b, x, Base::m_preconditioner, m_iterations, m_error);
		m_info = m_error <= Base::m_tolerance ? Success : NoConvergence;
	}

  protected:
};

} // end namespace Eigen

#endif // EIGEN_MINRES_H
